Thermodynamic Stability of Driven Open Systems andControl of Phase Separation by Electro-autocatalysis

Martin Z. BazantDepartments of Chemical Engineering and Mathematics,

Massachusetts Institute of Technology, Cambridge, MA 02139 USA andPresent address: Department of Materials Science and Engineering and SUNCAT

Interfacial Science and Catalysis, Stanford University, Stanford, CA 94305(Dated: April 4, 2017)

Motivated by the possibility of electrochemical control of phase separation, a variational theory ofthermodynamic stability is developed for driven reactive mixtures, based on a nonlinear generaliza-tion of the Cahn-Hilliard and Allen-Cahn equations. The Glansdorff-Prigogine stability criterion isextended for driving chemical work, based on variations of nonequilibrium Gibbs free energy. Linearstability is generally determined by the competition of chemical diffusion and driven autocatalysis.Novel features arise for electrochemical systems, related to controlled total current (galvanostaticoperation), concentration-dependent exchange current (Butler-Volmer kinetics), and negative differ-ential reaction resistance (Marcus kinetics). The theory shows how spinodal decomposition can becontrolled by solo-autocatalytic charge transfer, with only a single Faradaic reaction. Experimentalevidence is presented for intercalation and electrodeposition in rechargeable batteries, and furtherapplications are discussed in solid state ionics, electrovariable optics, electrochemical precipitation,and biological pattern formation.

I. INTRODUCTION

This Faraday Discussion[1] focuses on the use of elec-tric fields to control the dynamical response of materials,such as electroactuation of polymer gels and electrovari-able optics with plasmonic nanoparticles. Although ithas not been widely recognized, these phenomena couldbe strongly affected by phase separation of the con-stituents into domains of different density or chemicalidentity. Here we consider the possibility of controllingsuch phase separation by electrochemical reactions. Thisraises fundamental questions about thermodynamic sta-bility, which we motivate by first summarizing the phys-ical picture behind our results.

II. PHYSICAL PICTURE

A. Thermodynamic Stability Near Equilibrium

Consider a system containing a chemical species A atuniform concentration c, which is thermodynamically un-stable to concentration fluctuations. In particular, at-tractive inter-particle forces favor phase separation intostables phases of higher and lower concentration, whichcorrespond to local minima of the homogeneous Gibbsfree energy gh(c). As discussed below, Gibbs himself de-veloped the original stability criterion for chemical mix-ture near equilibrium:

Stable:d2ghdc2

=dµhdc

> 0 (1)

where µh(c) is the diffusional chemical potential of thehomogeneous mixture, defined as the change in free en-ergy upon adding a particle of species A at constant tem-perature and pressure.

The Gibbs criterion (1) has a simple graphical inter-pretation, shown in Fig. 1 for a binary mixture withtwo stable equilibrium states, corresponding to two lo-cal minima of gh(c) or zeros of µh(c) = g′h(c). In the“miscibility gap” between the minima, it is favorable tophase separate into a linear combination of the two sta-ble states having the same average concentration, whosefree energy lies on a common tangent construction. Thesame principle can be applied to small concentration fluc-tuations using a local secant construction, which showsthat stable concentrations correspond to a locally convexfree energy, g′′h(c) > 0, or increasing chemical potential,µ′h(c) > 0. Within the “chemical spinodal” where con-vexity is lost, g′′h(c) = µ′h(c) < 0, the system is unstableto spontaneous phase separation (“spinodal decomposi-tion”) [3].

B. Stability of Mixtures with Driven ChemicalReactions

The theory of thermodynamic stability has been ex-tended to include chemical reaction networks in closedbulk systems with porous boundaries [5], such as biolog-ical cells, but here we focus instead on driven chemicalreactions in open bulk systems. The basic principles areillustrated by driven adsorption,

Mres −→ M (2)

where a single species M evolves with local chemical po-tential µ(x, t) and undergoes homogeneous reactions witha reservoir at constant chemical potential, µres, whereit takes the form of (possibly different) species Mres.For bulk mixtures, this model could describe a reactivespecies M at low concentration in a sea of equilibratedmolecules, which includes the reaction product Mres, as

arX

iv:1

704.

0060

8v1

[ph

ysic

s.ch

em-p

h] 3

Apr

201

7

2

!c !c + Δ!c !c − Δ!c

µh

gh

(a)

(b)

!c

FIG. 1. Thermodynamic stability an inert, homogeneous bi-nary mixture (described by the regular solution model [2–4]).(a) Homogeneous free energy and (b) diffusional chemical po-tential versus dimensionless concentration, showing the com-mon tangent construction for phase separation in the misci-bility gap (red). The secant construction for linear instabilityin the chemical spinodal region (dashed blue) is shown in (a).

in open-system models of self-organization in biologicalcells [6].

The same model also describes a wide variety of ad-sorption phenomena at solid or liquid interfaces, suchas monolayer adsorption, where attractive lateral forcescan drive pattern formation [4]. This tendency for clus-tering modifies the classical theory of surface adsorp-tion [8, 9] and sorption hysteresis in porous media [10].Similar phenomena can occur for the solid-state insertionof bulk neutral species, such as hydrogen into palladiumhydride [11–13], or charged species at electrodes, such aslithium ions into iron phosphate [4, 7, 14, 15], shown inFig. 2.

A key result of our general stability analysis below isthat a fast driven reaction can suppress phase separationat constant potential µres if the reaction rate R decreaseswith reaction extent,

Stable:

(dR

dc

)µres

< 0 (constant potential) (3)

Electrochemical systems offer the unique capability ofcontrolling the rate of Faradaic relations, and this leadsto a new phenomena of phase separation at constant cur-rent. In the usual case of positive reaction resistance (de-

II0

= 0.01

Li+ + FePO4 + e− → LiFePO4

+

FePO4

LiFePO4

3.2

3.25

3.3

3.35

3.4

3.45

3.5

0 0.2 0.4 0.6 0.8 1

Batte

ry v

olta

ge (V

)

x in LixFePO4

I/I0=.001I/I0=.25

I/I0=.5I/I0=1I/I0=2

slow phase separation

fast uniform insertion

II0

= 0.033 II0

= 0.05

(a)

(b)

FIG. 2. Control of coherent phase separation in a binarysolid Li-ion battery cathode (LiXFePO4) by Faradaic inser-tion reactions. (a) Predicted battery voltage versus lithiummetal (V = V Θ − µ/e) and (b) surface lithium concentrationprofiles at X = 0.6 for different applied currents, scaled to areference exchange current, I0. [Adapted from Cogswell andBazant [7]]

fined below), phase separation is suppressed if the reser-voir potential increases with reaction extent:

Stable:

(dµresdc

)R

> 0 (constant current) (4)

which is a generalization of the Gibbs criterion (1) for achemically driven, open system. This effect is clearly seenin the lithium insertion simulations of Fig. 2, where thebattery voltage becomes monotonically decreasing withconcentration (dµdc = −edVdc > 0), as concentration fluc-tuations disappear above a critical current.

In summary, phase separation is reduced if the reactionis auto-inhibitory (either slows down or becomes harderto drive), or enhanced if it is auto-catalytic (either accel-erates or becomes easier to drive).

C. Solo-autocatalysis

We refer to this nonlinearity for a single reactionin a concentrated mixture as “solo-autocatalysis” to

3

distinguish it from the traditional concept of “collec-tive autocatalysis” for chemical reaction networks in di-lute mixtures, governed by mass action kinetics. Solo-autocatalysis is an inescapable feature of adsorption, in-tercalation and deposition reactions. Whenever the prod-uct (or reactant) occupies a finite set of sites, it neces-sarily affects the subsequent reaction rate. Adsorptionreactions are typically solo-autoinhibitory (rate suppress-ing) at high concentration, as product covers the activesites. Since the reaction creates a particle M while de-stroying a vacancy V, the vacancy can be viewed as anadsorption catalyst, Mres + V −→ M, which slowly dis-appears as the reaction progresses. Vacancies can also beviewed as a distinct chemical species in a reactive binarymixture with the adsorbed particles. The total volumeconstraint yields a single concentration variable, cM , cV ,or dimensionless coverage, c = cM/cs (where cs = siteconcentration), which evolves in response to differencesin “diffusional chemical potential”, µ = µM − µV , eitherby diffusion or reactions [3, 4, 16, 17]. The same ap-plies to the isomerization reaction, M −→ V, in a closedsystem [18], which corresponds to µres = 0.

In general, it may not be possible to identify vacanciesor other catalytic species, and yet the reaction rate stilldepends on concentration. In particular, electrochemicalreactions tend to be solo-autocatalytic (rate enhancing)at low concentration, as redox active molecules increasethe exchange rate for electron transfer [4, 19], while re-maining auto-inhibitory at high concentration. The re-sult is a “volcano” shaped exchange current versus con-centration, which is usually assumed to be symmetric,I0 ∼

√c(1− c), in models of Li-ion batteries [20, 21]

and fuel cells [22–24]. In contrast, the theory of chargetransfer based on non-equilibrium thermodynamics pre-dicts an asymmetric exchange-current volcano favoringhigher rates at low concentrations, considering only siteexclusion in the transition state [4]. As we shall see, thisturns out to be the key property that enables the controlof phase separation [7, 14, 15].

D. Control of Phase Separation byElectro-autocatalysis

The fundamental mechanism for control of phase sep-aration by driven autocatalysis is illustrated in Fig. 3,in the case of anodic ion insertion, or adsorption of aneutral species, at constant current. The externally con-trolled potential µres is equal to the internal potentialµh(c) (for a homogeneous base state) plus the affinity,A = µres − µh(c), which controls the reaction rate. Inthe case of Faradaic reactions transferring n electrons,it is the (anodic) activation overpotential, η = A/ne,that controls the Faradaic (oxidation) current, I = nev.The simplest autocatalytic model has a separable form,I = I0(c)f(neη/kBT ), with a concentration-dependentexchange current, I0(c) and monotonic overpotential de-pendence (f ′ > 0, f ′(0) = 1, f(0) = 0), as in the Bulter-

Volmer equation [19] and various generalizations for con-centrated solutions [4], considered below.

The reservoir chemical potential, or cell voltage V , thusdepends on concentration and the applied current,

µres = V − V0 = µh(c) + f−1

(I

I0(c)

)∼ µh(c) +

I

I0(c)(5)

where potential is scaled to kBT and voltage to kBT/ne,V0 is the open circuit voltage at µ = 0, and, for clar-ity, we linearize the overpotential dependence – but notthe autocatalytic concentration dependence. As shownin Fig. 3(a)), for a non-autocatalytic reaction (I ′0 = 0),the activation overpotential is constant, so the shape ofthe voltage profile and stability of the system cannot bealtered by the reaction.

Autocatalysis is required to alter thermodynamic sta-bility. As shown in Fig. 3(b)), for concentrations wherethe insertion reaction is auto-inhibitory (I ′0(c) < 0), thehomogeneous state becomes stable (µ′res > 0) above acritical current, even within the spinodal region, whichamounts to electrochemical freezing of a thermodynam-ically unstable mixture in a disordered state. The sys-tem’s entropy is increased above its equilibrium value byapplying external work to drive the reaction. This phe-nomenon is different from rapid quenching of a liquid toa metastable glass or amorphous solid, because the elec-trochemically frozen mixture is stable under the appliedcurrent. As soon as the current drops below the criticalvalue, however, spontaneous phase separation occurs.

Interestingly, when the current is reversed, the oppo-site phenomena occur. Phase separation is enhanced inthe spinodal region, and the homogeneous mixture out-side the spinodal can be destabilized. The latter cor-responds to electrochemical melting of a thermodynami-cally stable disordered state to form two ordered phases.Again this is not a transient phenomenon, but a changeof thermodynamic stability in which the external workdriving the reaction makes it favorable to lower of thesystem’s entropy.

As shown in Fig. 3(c)), for concentrations where thereaction is autocatalytic (I ′0(c) > 0), the system becomesmore unstable with increasing insertion current. Abovea critical insertion current, phase separation can occuroutside the spinodal region, which corresponds to electro-chemical melting of a thermodynamically stable mixture.Conversely, extraction currents now stabilize the systemand can lead to electrochemical freezing of the spindoalregion below a threshold negative current.

In summary, the theory predicts the following effectsof electro-autocatalysis on phase separation at constantcurrent:

• During periods of auto-inhibition (I ′0(c) < 0), theforward reaction (I > 0) suppresses phase separa-tion (completely for I > Ic), while the backwardreaction (I < 0) enhances it.

• During periods of autocatalysis (I ′0 > 0), the for-ward reaction (I > 0) enhances phase separation,

4

(a) (b) (c)

!I0

!V

!c !c !c

stable

unstable

unstable

unstable

stable

unstable

constant exchange rate

auto-inhibitory auto-

catal

ytic

auto-catalytic auto-

inhibi

tory

FIG. 3. Principles of thermodynamic stability controlled by electro-autocatalysis. Top row: Dimensionless exchange current vs.product concentration I0(c). Bottom row: Dimensionless electrode voltage versus concentration at different applied currents forinsertion (red) and extraction (blue), where signs are chosen for anodic cation insertion to resemble neutral-species adsorption

(V = µres). (a) A non-autocatalytic reaction (I ′0 = 0) simply shifts the potential curves up and down by constant activationpotential, and thus cannot alter the spinodal region of instability (negative slope, between dashed lines). (b) An auto-inhibitory

reaction (I ′0 < 0) in the spinodal reaction can suppress the instability (positive slope) leading to “electrochemical freezing”above a critical insertion current, while further destabilizing the system during extraction. Outside the spinodal, the reactioncreates instability and leads to “electrochemical melting” above a critical current, while further stabilizing the mixture duringextraction. (c) An auto-catalytic reaction in the spinodal region (I ′0 > 0) has the opposite effect of destabilization duringinsertion and stabilization during extraction.

while the backward reaction (I < 0) suppresses it.

These predictions have recently been verified in experi-ments on Li-ion battery materials, as discussed below inSection VI.

E. Nonequilibrium Gibbs Free Energy

In the examples above, the applied current appears toact as an independent state variable, analogous to tem-perature, pressure and concentration. In hindsight, thereason is that constant current contributes a well-definedstate-dependent excess energy (the activation overpo-tential) to the total non-equilibrium Gibbs free energy,G(c, I) of the driven open system. Comparing Eqs. (1)and (14), such a state function could be defined as

∆G(c, I) =

∫ c

c0

µres(c, I) dc = ∆Geq(c) + ∆iWd(c, I) (6)

where we define the reversible change in equilibrium freeenergy, associated with the transformation at zero cur-rent,

∆Geq(c) = ∆G(c, 0) = gh(c)− gh(c0) (7)

and the irreversible driving work done on the system atfinite current,

∆iWd(c, I) =

∫ c

c0

A(c, I) dc =

∫ t

t0

I2RF (c, I)dt (8)

For Faradaic reactions, the latter is equal to the time-integral of the electrical power, Pe = I2RF , whereRF = −η/I > 0 is the Faradaic resistance. This simpleexample will help us generalize the theory of thermody-namic stability for driven open systems.

F. Driven Autocatalysis versus Chemical Diffusion

The preceding simple analysis considers driven auto-catalytic reactions which are fast compared with diffu-sion (large Damkohler number, Da > 1, defined below).In the opposite limit of negligible reactions, Cahn pio-neered the theory of diffusion-driven spinodal decomposi-tion [25–27]. The instability is controlled by the chemicaldiffusivity [3],

D =D c

kBT

dµhdc

(9)

5

which enters Fick’s law (flux = −D∇c) for a concen-trated solution, where D > 0 is the tracer diffusivity inthe dilute limit [4]. Outside the spinodal, “forward dif-fusion” (D > 0) leads to familiar smooth concentrationprofiles, but inside the spinodal, the system is destabi-lized by “backward diffusion” (D < 0) leading to phaseseparation.

Here, we show that thermodynamic stability of reac-tive mixtures is determined by the competition of auto-catalysis and chemical diffusion. In driven open systems,such as electrochemical interfaces, this competition canbe controlled by applied potentials and currents. Thetheory predicts that stable equilibrium mixtures can bedriven to form desired patterns by electrochemical melt-ing, while unstable mixtures can be driven to remain ho-mogeneous by electrochemical freezing. These surpris-ing phenomena not only have applications to electroac-tuation, but they also raise profound questions aboutnonequilibrium thermodynamics, as we now explain.

III. BACKGROUND

In order to analyze the stability of driven open systems,we must first extend nonequilibrium chemical thermody-namics [5, 28] for inhomogeneous systems, as describedby phase-field models [3, 29], using the calculus of varia-tions [30].

A. Gibbs’ Stability Theory for Inert Mixtures

Gibbs pioneered the theory of thermodynamic stabil-ity [31], based on the notion that entropy is maximizedin equilibrium [5]. As such, any perturbation of a stableequilibrium must lower its entropy (or increase its free en-ergy) according to ∆S = S−Seq = δS+ 1

2δ2S+. . ., where

the first and second variations of the entropy functionalwith respect to spatial perturbations in concentration,temperature, etc. must satisfy

Stable equilibrium: δS = 0 and δ2S < 0. (10)

For fluctuations in temperature or volume, the Gibbs sta-bility criterion implies positive heat capacity, Cv > 0, andisothermal compressibility, κT > 0.

For concentration fluctuations δci at constant inter-nal energy and volume, stable equilibrium requires [5]

δ2S = −∫V

∑i,j

δci

(δ

δcj

µiT

)δcjdV < 0 (11)

where we define the (diffusional) chemical potential

µi =δG

δci(12)

as the first variational derivative of the Gibbs free en-ergy with respect to the concentration of species i. This

is the continuum analog of the familiar definition from

statistical mechanics, µi =(

∆G∆Ni

)T,P

, as the change in

free energy from adding a particle of species i, where a“particle” corresponds to a Dirac delta function added tothe concentration profile at a given position [4].

With this generalization, Gibbs’ maximum entropycondition, Eq. (11), implies that the Hessian tensor ofsecond variational derivatives, G′′, must be positive def-inite in equilibrium,

δµiδcj

=δ2G

δciδcj= G′′ij > 0 (13)

(We write Tij > 0 if∑ij

∫VδuiTijδuj dV > 0 for all δui,

δuj .) In the limit of long-wavelength fluctuations in auniform system, this asserts that the homogeneous freeenergy density, gh(ci), has a positive definite Hessianmatrix of second partial concentration derivatives,

G′′ij =∂2gh∂ci ∂cj

> 0 (14)

In order words, in stable equilibrium, the free energymust be locally convex with respect to concentration,as shown in Fig. 1. The variational formula, Eq. (13),extends this concept to nonuniform systems and arisesnaturally in our nonequilibrium stability analysis below.

B. Thermodynamics of Inhomogeneous Systems

In contrast to classical thermodynamic models [5], weallow the Gibbs free energy functional, G[ci], to haveexplicit dependence on concentration gradients, whichcould arise from interfacial tension, elastic coherencystrain, electrostatic energy, or other non-idealities of in-homogeneous systems. In Eqs. (12) and (13), we intro-duce notation for the first, second, and higher variationalderivatives,

∆G = δG+1

2δ2G+ . . . (15)

=

∫V

∑i

δci

δGδci

+∑j

δcj

(1

2

δ2G

δciδcj+ . . .

) dV

defined by the expansion of the free energy change inresponse to one, two or more simultaneous bulk concen-tration fluctuations (which vanish on the boundary), re-spectively.

In order to describe the dynamics of phase separation,it is necessary to model interfacial tension between phaseswithout artificially introducing sharp phase boundaries.In 1893, Van der Waals first proposed adding a quadraticgradient penalty to the homogeneous free energy [32, 33],

G[c] =

∫V

(µΘc+ gh(c) +

K

2|∇c|2

)dV (16)

6

where we include a reference chemical potential [4], µΘ.The gradient penalty term, K

2 |∇c|2 = κ

2 |∇c|2, is often

written in terms of filling fraction, c = c/cs, over sitesof density cs, where κ = Kc2s, and can be adjusted tofit the tension and thickness of phase boundaries. Thisvisionary idea was somehow forgotten for over half acentury, until its rediscovery in physics by Landau andGinzburg [34] (to describe magnetic flux in type II su-perconductors) and in materials science by Cahn andHilliard [2] (to describe phase separation in solid binaryalloys).

Led by Cahn [25, 26, 35–39], this approach paved theway for modern phase-field models [3, 29], which approxi-mate phase boundaries as localized, but continuous, “dif-fuse interfaces”. Taking a functional derivative of Eq.(18), the diffusional chemical potential (per site) µ andits homogeneous limit µh are given by

µ = µh −K∇2c and µh = µΘ +dghdc

. (17)

Equilibrium concentration profiles satisfy the Beltramiequation, µ = δG

δc = constant. Solutions in the misci-bility gap describe uniform stable domains separated bydiffuse phase boundaries, whose width, λ =

√κ/csΩ, and

interfacial tension, γ =√κcsΩ, are related to the gradi-

ent penalty κ and a characteristic energy barrier betweenthe stable concentrations, Ω, e.g. the regular solution pa-rameter for pairwise interatomic forces [2, 3].

For multicomponent, anisotropic, inhomogeneous sys-tems, the Cahn-Hilliard free energy, chemical potentials,and Hessian tensor are given by

G =

∫V

∑i

µΘi ci + gh(ci) +

1

2

∑ij

∇ci ·Kij∇cj

dV(18)

µi = µΘi +

∂gh∂ci−∑j

∇ ·Kij∇cj (19)

δµiδcj

=∂2gh∂ci∂cj

+∇δciδci

·Kij∇δcjδcj

(20)

where the Hessian depends on gradients of the fluctua-tions, according to Eq. (16).

C. Linear Irreversible Thermodynamics ofDiffusion

Gradients in chemical potential provide thermody-namic forces that drive diffusional fluxes, respectively,

Fi = −∇µiT

and Ji =∑j

LijFj (21)

where we make the ubiquitous approximation of LinearIrreversible Thermodynamics (LIT) [5], which is validclose to local equilibrium. The linear response matrixmust be symmetric, Lij = Lji (Onsager relation), and

positive definite, in order to ensure a positive entropyproduction rate by diffusion,

diS

dt=

∫V

(∑i

FiJi

)dV =

∫V

∑ij

FiLijFj

dV > 0

(22)Mass conservation with LIT fluxes yields the (multi-component) Cahn-Hilliard equation,

∂ci∂t

= ∇ ·∑j

Lij∇δG

δcj(23)

which is the standard model for phase separation by dif-fusion in a closed system [3, 29], including linear insta-bility and spinodal decomposition [25, 26, 37]. The On-sager coefficients are related to the mobility tensor (driftvelocity per force) via Lij = Mijcj . For a single diffusingspecies, the tracer diffusivity satisfies the Einstein rela-tion, D(c) = M(c)kBT , and takes the form D = D0(1−c)or L ∼ c(1 − c) in a binary mixture [16], to reflect thecrowding of sites [4].

The phase-field LIT formalism can be extended to elec-trochemical systems [4], which have long-range Coulombforces in addition to the short-range forces that deter-mine gh. The electrochemical potential is defined byadding the electrostatic energy qiφ to µ, and the asso-ciated Nernst-Planck LIT flux (ionic current) includescontributions from diffusion and electromigration. Themobility matrix Lij is usually assumed to be diagonal,but this neglects strongly coupled fluxes at high concen-trations, where strong Coulomb correlations may yieldnegative off-diagonal coefficients [40]. The electrostaticpotential of mean force, φ, is determined either by elec-troneutrality or Poisson’s equation.

D. Prigogine’s Stability Theory for ReactiveMixtures

Let us now consider the effect of chemical reactions,Mr,m =

∑i sr,m,iMr,m,i ←→

∑j sp,m,jMp,m,j = Mp,m,

where Mr,m and Mp,m are the reactant and product com-plexes of the mth reaction with total chemical potentials,µr,m =

∑i sr,m,iµr,m,i and µp,m =

∑j sp,m,jµp,m,j , and

stoichiometric coefficients, sr,m,i and sp,m,j, respec-tively. For electrochemical reactions, the chemical speciesMi include both ions and electrons. The thermody-namic driving force for a reaction is the change in Gibbsfree energy [5, 41], ∆rGm = µp,m−µr,m, which is equal tothe difference in diffusional chemical potentials [4]. For aFaradaic reduction reaction transferring n electrons, theactivation overpotential, ηm = ∆rGm/ne, is the free en-ergy of the net reduction reaction per charge [4].

De Donder pioneered non-equilibrium chemical ther-modynamics and related the free energy of reaction to

7

the chemical affinity [42, 43],

Am = −(∂G

∂ξm

)T,P

= −∆rGm (24)

where G is the total Gibbs free energy, including reac-tants and products, and ξm is the extent of the reac-tion. He also argued that the free energy of reactioncontributes to Clausius’ “uncompensated heat”, dQ′ (orirreversible entropy production, diS, in modern terminol-ogy [42, 44]) and introduced the equivalent definition,

Am =

(dQ′

dξm

)P

= T

(diS

dξm

)P

= µr,m − µp,m = −ne ηm(25)

where we also relate affinity to activation overpotentialof a reduction reaction [4].

The affinity can be viewed as a thermodynamic force,Fm = Am

T , whose conjugate thermodynamic flux, Jm =Rm, is the reaction rate

Rm =1

V

dξmdt

= −∑i

sr,m,idcidt

=∑j

sp,m,jdcjdt

(26)

(In thermodynamics [5, 41, 45, 46], this is “reaction ve-locity”, vm, but we adopt our previous notation for “reac-tion rate” [4], Rm, which also avoids any confusion withfluid velocity in liquid systems!) For thermodynamic con-sistency, the reaction rate must satisfy only two funda-mental constraints:

1. Equilibrium must correspond to detailed balance ofthe forward and backward rates

Am = 0 ⇔ Rm = 0. (27)

2. Out of equilibrium, the net reaction must proceedin the direction of the affinity, which De Donderwrote expressed as positive irreversible entropy pro-duction per volume [42],

σm = AmRm = −ηmImV

> 0. (28)

For Faradaic reactions, the integral reaction resis-tance must be positive, Ri = −ηm/Im > 0, al-

though the differential resistance,Rd = −dηmdIm, may

have either sign, as discussed below.

Prigogine [41, 47] showed that closed reaction network isstable if the affinities decrease with each reaction extent,

Stable:

(δAmδξn

)P

= T

(δ2i S

δξmδξn

)P

< 0. (29)

or equivalently that the irreversible entropy reaches amaximum in equilibrium, which follows from Gibbs’ max-imum entropy principle, Eq. (10), and De Donder’s def-inition of affinity, Eq. (25).

E. Linear Irreversible Thermodynamics ofReactions

For a closed system in equilibrium, the irreversible en-tropy production vanishes. Close to equilibrium whereLIT applies, Prigogine [47] showed that the entropy pro-duction rate Pe decreases and reaches a local minimumfor any stationary non-equilibrium state[5, 41, 48],

Pe =diS

dt=

∫V

(∑α

FαJα

)dV > 0, Stable:

dPedt

< 0

(30)where the sum is over all pairs of conjugate forces Fα andfluxes Jα, including each affinity and reaction rate. Theentropy production rate acts as a Lyapunov functional(Pe > 0, Pe < 0), which can also determine the stabilityof non-equilibrium states [5, 48].

The analog of LIT fluxes for chemical reactions is theassumption of linear kinetics, which we express variation-ally as

Rm = kmAm = km∑i

sm,iδG

δci(31)

where km > 0 is a constant and sm,j = sp,m,j − sr,m,j(positive stoichiometric coefficients for products, nega-tive for reactants). Although widely used, linear kinet-ics are strictly only valid near equilibrium in dilute mix-tures [4, 49]. Mass conservation equations take the form,

∂ci∂t

=∑j

(∑m

kmsm,ism,j

)δG

δcj(32)

for a closed chemical reaction network.In this work, we focus instead on chemical reactions

in open systems. The standard phase-field model for adriven open system is the Allen-Cahn equation [3, 38],

∂c

∂t= kresAres = kres

(µres −

δG

δc

)(33)

where µres is the chemical potential of an external reser-voir of species c. The Allen-Cahn equation is usuallyapplied to non-conserved order parameters, such as thedegree of solid-like order in liquid solidification, but whenapplied to chemical reactions, it corresponds to linear ki-netics for a driven reaction. As a result of this assump-tion, we shall see that the Allen-Cahn equation predictsthe same spinodal region for a driven open systems asfor closed equilibrium systems, Eq. (14), as shown inFig. 3(a). This is true even when diffusion is includedin a combined Cahn-Hilliard/Allen-Cahn model [18]. Asrecognized by Prigogine, nonlinear thermodynamics arerequired for any departures from the equilibrium “ther-modynamic branch” of stability [5, 44–46], and we shallsee that this also holds true for the stability of drivenopen systems, such as electrochemical cells.

8

F. Nonlinear Irreversible Thermodynamics ofReactions

Huberman [50] added mass-action kinetics to theCahn-Hilliard equation as a model for spinodal decom-position and pattern formation in a reactive mixture,

∂c

∂t= ∇ · L∇δG

δc+R(c). (34)

Similar Ginzburg-Landau-type reaction-diffusion equa-tions have been studied extensively in chemical physicsas generic models of self-organization [51]. Glotzer etal. performed simulations and linear stability analysisof Eq. (34) and reached the tantalizing conclusion thatreactions could be used to alter the spinodal region andcontrol pattern formation. However, Lefever et al. [52]pointed out that the model is not thermodynamicallyconsistent, since equilibrium (µ =constant) is neither sta-tionary (∂c∂t = 0) nor in detailed balance (f = 0), andequilibrium states depend on the mobility or diffusivity.Instead, the reaction rate must satisfy the two constraintsgiven above, Eqs. (27)-(28), and the further assumptionof linear kinetics (31) eliminates any effect on the spin-odal region.

A thermodynamically consistent linear stability analy-sis for general chemical reaction networks was performedby Carati and Lefever [17], based on multi-componentCahn-Hilliard diffusion (23) and (17) and a nonlinear re-action model converting species i into species j:

Rij = fr(µi)− fr(µj), f ′r > 0 (35)

which upholds Eqs. (27)-(28). Notably, they also con-sidered open reaction networks with chemostats and pre-dicted the possibility of “chemical freezing” of phase sep-aration, by two or more collectively autocatalytic reac-tions.

Hildebrand, Mikhailov and Ertl [53] analyzed generalstochastic models of surface adsorption and also con-cluded that “thermal adsorption and desorption pro-cesses do not prevent macroscopic phase separation”, but“if, on the other hand, an energetically activated process(such as photo-desorption) is present, kinetic freezing ofphase separation, leading to the formation of stationarynonequilibrium structures, can occur,” consistent withexperiments and simulations on reactive monolayers [54].In other words, the reaction must be driven, supplyingexternal work. Here, we focus on the possibility of usingFaradaic reactions as the driving process.

G. Variational Electrochemical Kinetics

We shall modify some of these conclusions using moregeneral models, based on transition-state theory for con-centrated solutions and electrochemical systems [4]. Thetheory is based on variational definitions of activity,

ai = γici, activity coefficient γi, and excess chemical po-tential, µexi = kBT ln γi:

µi =δG

δci= µΘ

i + kBT ln ai = kBT ln c+ µexi (36)

For the reaction, Mi → Mj , the generalized Eyring rateis given by

Rij = k0

(e−(µex

‡ −µi)/kBT − e−(µex‡ −µj)/kBT

)=k0(KΘ

ijai − aj)γ‡

(37)where KΘ

ij is the equilibrium constant and γ‡ is the ac-tivity coefficient of the transition state, which generallydepends on concentration, e.g. γ−1

‡ = (1 −∑l cl)

s for sexcluded sites on a lattice.

As a result, the model is more general than Eq. (35)

and allows for negative differential resistance (∂Rij

∂µi< 0),

as in Marcus kinetics, and solo-autocatalysis (∂Rij

∂ci6= 0).

The latter includes the important case of Butler-Volmerkinetics [4]:

I = neR = I0

(e−αη − e(1−α)η

), η =

neη

kBT(38)

I0 =nek0(aOae)

1−αaαRγ‡

(39)

for the reduction reaction, O + ne− → R.Once the reaction model is specified, the thermody-

namically consistent set of reaction-diffusion equationstakes the form [4],

∂ci∂t

= ∇ ·∑j

Lij∇δG

δcj+∑m

si,mRm

(ci,

δG

δci

)(40)

where we write the mth reaction as ∅ →∑i si,mMi. This

is the most general mathematical framework for concen-tration evolution, based on LIT fluxes and nonlinear ir-reversible thermodynamics for chemical reactions.

H. Glansdorff-Prigogine Nonequilibrium StabilityTheory

Glansdorff and Prigogine derived a general linear sta-bility condition for stationary non-equilibrium states ofreactive mixtures far from equilibrium [5, 46, 55], basedon variations of irreversible entropy production [45, 56].They argued that the second variation of the entropy actsas a Lyapunov functional, L = − 1

2δ2S, which measures

the “distance” from a stationary state, L > 0, and thusdecreases with time if it is stable, dL

dt < 0. The stabil-ity criterion can be expressed as a constraint of positiveexcess entropy production [46, 55],

Stable:d

dt

δ2S

2=

∫V

(∑α

δFα · δJα

)dV > 0, (41)

9

ΔdS = − ΔWd

TΔiS > 0ΔeS

FIG. 4. Three contributions to entropy production in adriven open system: (1) bulk irreversible entropy production,∆iS, (2) entropy flow due to mass and energy flow throughthe boundary, ∆eS, and (3) driven entropy production, ∆dS,due to the work, ∆Wd, done on the system by exchangingmass and energy directly between the external reservoirs andthe interior bulk. The image shows a two-phase lithium ironphosphate nanoparticle driven far from equilibrium by an ap-plied Faradaic current [15] from Fig. 7 below.

for an arbitrary set of conjugate forces Fα and fluxes Jα.Besides reactions, there may also be contributions to ex-cess entropy production from diffusion, electromigration,elastic deformation, heat conduction, etc. The same re-sult holds for any boundary conditions in which eitherthe forces or fluxes are held fixed, causing the secondvariation of the entropy flow to vanish on the boundary.

Although the Glansdorff-Prigogine criterion (41) fol-lows from thermodynamically consistent mass and energybalances [5], Keizer and Fox first expressed “qualms”about its validity [57] and triggered a long debate [58–61]. They provided counter-examples of auto-catalyticreaction networks in dilute solutions [57, 61], whose non-equilibrium steady states violate Eq. (41), and yet couldbe described by Keizer’s stochastic thermodynamics [62–65]. Glansdorff, Nicolis and Prigogine responded thatdifferent Lyapunov functions are possible depending onthe choice of conservation laws [58, 60], and pointedto Schlogl’s earlier derivation of Eq. (41) based onsimilar stochastic principles [66], rooted in fluctuation-dissipation theorems for nonequilibrium states [67].

We shall see that the problem has to do with driven,open systems. In the counter-examples, nonequilbriumstationary states are constructed by fixing certain con-centrations or production rates throughout the domain,but such “chemostats” are neglected in the Glansdorff-Prigogine derivation [5], which assumes an unconstrainedsystem of reaction-diffusion conservation laws. The sta-bility criterion cannot be expressed in terms of affinitiesby summing over all reactions, if any concentrations orrates are externally controlled.

As shown in Fig. 4, the theory generally does not ac-count for bulk entropy flow from distributed work doneby “active matter” or by the direct exchange of massand energy with external reservoirs. Moreover, the tra-ditional focus on reaction networks in dilute solutionsobscures the rich new physics of driven reactions coupled

with phase transformations. Here, we generalize the the-ory for concentrated solutions and show how driven re-actions can control thermodynamic stability.

IV. THEORY

A. Wisdom from Stochastic Thermodynamics

Any theory of nonequilibrium thermodynamics forconcentrated systems should be consistent with stochas-tic thermodynamics for the ideal limit of a dilute systemwith chemical reaction networks obeying mass action ki-netics [65, 68–72]. Rao and Esposito recently summa-rized this “wisdom” and rigorously defined various formsof the non-equilibrium Gibbs free energy for open systemswith driven chemical reactions [72],

G = Geq + kBTL (42)

where Geq is the local equilibrium free energy of a statethat would reached if the external driving were stoppedand the system were allowed to relax under the imposedconstraints and kBL is the “relative entropy” betweenthe equilibrium and nonequilibrium states. The relativeentropy, also known as the Kullback-Leibler divergencein information theory [73], is a non-negative measure ofthe “information gain” between two probability distribu-tions, which acts as a Lyapunov functional for the relax-ation to local equilibrium.

The change in non-equilibrium free energy between twostates,

∆G = ∆Wd − T∆iS (43)

has contributions from external work and internal en-tropy production of opposite sign. The work can bebroken into irreversible and reversible parts, ∆Wd =∆iWd+∆Geq, where the reversible chemical work is equalto the change in local equilibrium free energy, as a resultof exchanging bulk particles with the reservoirs. Com-bining these equations, we arrive at the central result ofRao and Esposito for irreversible chemical work in dilutemixtures [72],

∆iWd = ∆Wd −∆Geq = kBT∆L+ T∆iS (44)

The Second Law (∆iS > 0) then implies a “non-equilibrium Landauer principle” [72], ∆iWd ≥ kBT∆L,which provides a lower bound on the irreversible exter-nal work associated with the fluctuation (the thermody-namic cost of information gain [74, 75]) that vanishes fortransitions between equilibrium states (∆L = 0).

B. Thermodynamic Stability of Driven, OpenSystems

Let us apply these principles more broadly to concen-trated systems experiencing arbitrary forms of external

10

driving work. Enthalpy from inter-particle forces nowleads to nonlinear chemical diffusion and influences theenthalpies of reactions, both internal and external. Asa result, the equilibrium free energy may lose convex-ity and lead to spinodal decomposition. Thermodynamicstability will then be influenced by the driving work ∆Wd

done on the bulk system, which may include contribu-tions from heat transfer (e.g. radiation), mass transfer(e.g. chemical reactions with reservoirs), external forces(e.g. magnetic fields or mechanical work), or internalenergy sources (e.g. swimming particles or other activematter).

These contributions are neglected in the prevailingtheory of nonequliibrium thermodynamics [5]. Pri-gogine and collaborators described the thermodynam-ics of closed, internal reaction networks in what couldbe termed “partially open” systems, in which entropyor energy exchange with external reservoirs occurs onlythrough the boundaries. In contrast, we consider “fullyopen” driven systems, in which entropy flow and externalwork can also be distributed across the bulk system.

The key theoretical concept is the nonequilibrium freeenergy, G. In some cases, it may be possible to con-struct G as a local state function in space and time,which depends on traditional intrinsic variables, such aschemical concentration, density, pressure and tempera-ture, as well as intrinsic external driving forces or fluxes.We have already discussed examples from the stochasticthermodynamics of chemical reaction networks [68, 72].Nonequilibrium free energies have also been constructedfor active suspensions of swimming particles [76–79] andrecently connected with stochastic thermodynamics [80].For driven electrochemical systems, we have already con-structed G(c, I) for ion adsorption in a phase-separatingelectrode at constant concentration c and constant cur-rent I in Eqs. (6)-(8). Below, we shall explicitly con-struct the nonequilibrium free energy (via its variationalderivatives) for a general homogeneous driven, open sys-tem.

In most cases, it is not possible to express G as a sim-ple state function due to various non-local, nonlinear pro-cesses in space and time, but we can still define the firstvariation of G, the response to an arbitrary fluctuation,as

δG ≡ δWd − TδiS = δGeq + δiWd − TδiS (45)

which can be integrated in time to obtain at least apath-dependent free energy, G(t). We can then identifya nonequilibrium steady state via

Steady State: δG = 0 ⇒ δiS =δWd

T(46)

which extends Gibbs’ condition of thermal equilibrium,δiS = 0, to account for driving work. The canonicalexample is a driven reaction network in detailed balance.We can also write the steady state condition as δStot = 0,where Stot = Si + Sd, where we define the change in

driving entropy

δSd = −δWd

T(47)

associated with the external reservoirs, which has theopposite sign of the driving work, since work creates orderand lowers entropy.

In the thermodynamic limit of a continuous system, lo-cal fluxes and reactions maintain each infinitesimal bulkvolume in quasi-equilibrium, leading to our first princi-ple:

Local equilibrium: δ2G = δ2Wd − Tδ2i S > 0 (48)

which generalizes Gibbs’ maximum entropy condition,Eq. (10), to account for driving work. The local equilib-rium condition can then be viewed as the Gibbs’ criterionfor the total entropy, δ2Stot < 0.

The definite sign of δ2G allows the second variationof nonequilibrium free energy to serve as a Lyapunovfunctional, which implies thermodynamic stability if itdecreases toward steady state (δG = 0) in response tofluctuations,

Stable:d

dtδ2G =

d

dtδ2Wd − T

d

dtδ2i S < 0 (49)

or ddtδ

2Stot > 0, which generalizes the Glandorff-Prigogine criterion of positive excess entropy production,Eq. (41), to account for excess driving entropy produc-tion, d

dtδ2Sd, or excess driving power, d

dtδ2Wd. Near

equilibrium, this also generalizes Prigogine’s principle ofminimum entropy production, Eq. (30).

The general stability criterion, Eq. (49), states thatthe excess entropy production from internal irreversibleprocesses must exceed the excess driving power. Eachterm can take either sign. If the excess driving poweris negative, it is possible to stabilize an ordered “dissi-pative structure” having negative excess entropy produc-tion [5, 42, 48]. Conversely, an unstable system can bedestabilized“chemically frozen” in a disordered state bypositive excess driving power. These surprising phenom-ena appear to contradict the Duhem-Jougeut Theorem,which asserts that a system that is stable to diffusionis also stable to chemical reactions [5, 41], but that isonly true in a partially open system without bulk drivingwork, under conditions derived below. Different behavioris possible in fully open, driven systems.

C. Variational Linear Stability Analysis

In order to illustrate these principles, we now per-form linear stability analysis on the most general ther-modynamically consistent system of isothermal reaction-diffusion equations, Eq. (40), using the calculus of varia-tions. For any concentration fluctuations δci around a

11

non-equilibrium base state, the simplest Lyapunov func-tion is the L2-norm of the perturbation,

Lc =1

2

∑i

∫V

(δci)2 dV ≥ 0 (50)

which must decrease for a stable base state,

Stable:dLcdt

=∑i

∫V

(∇δci · δJi +

∑n

si,nδciδRn

)dV < 0

(51)where we use the divergence theorem and assume δci = 0on the boundary. The general linear stability result (51)resembles the Glansdorff-Prigogine criterion (41) since itcontains products of excess thermodynamic fluxes (δJi,δRn) and certain excess forces, but the latter are ex-pressed in terms of concentration fluctuations (∇δci, δci),rather than fluctuations in proper thermodynamic forces(δ∇µi, δAn), which can only be derived for mass andenergy balances in partially open systems [5].

Assuming LIT fluxes and nonlinear reactions, Eq. (40),we can express the stability criterion as

dLcdt

=∑ij

∫V

[−∇δci ·

∑l

δcl

(∂Lij∂cl∇µj + Lij∇

δµjδcl

)+δciAijδcj − (∇δci)Dij(∇δcj)] dV < 0 (52)

where the first term involves fluctuations in the Onsagermatrix,

Lij =DijcikBT

(53)

and only applies to inhomogeneous base states with∇µj 6= 0. The second term also vanishes for a homo-geneous base state. The remaining terms comprise a dif-ference of two quadratic forms, whose physical meaningswe now explain.

D. Autocatalytic Rate and Chemical DiffusionTensors

For slow diffusion, the stability of a homogeneous basestate requires that the following tensor be negative defi-nite:

Aij =∑n

si,nδRnδcj

=∑n

si,n

(∂Rn∂cj

+∑l

∂Rn∂µl

δµlδcj

)< 0

(54)We refer to A as the “autocatalytic rate tensor”, sinceit describes how reaction rates depend on the extents ofboth products (si,n > 0) and reactants (si,n < 0) withinthe system, excluding all reservoir species. For linearstability with slow diffusion, a driven chemical reactionnetwork must be auto-inhibitory, A < 0. Prigogine’s sta-bility criterion based on affinities (29) follows in the case

of a closed system with linear kinetics (31), but Equation(54) based on reaction rates is much more general.

For slow reactions, a homogeneous base state is stableif the “chemical diffusion tensor” is positive definite:

Dij =∑l

Lilδµlδcj

> 0 (55)

Since the Onsager tensor, Lij , and the tracer diffusiontensor, Dij , are symmetric and positive definite, thisimplies that the Hessian tensor (13) must also be pos-itive definite. Therefore, the requirement of positive def-inite chemical diffusion tensor, Eq. (55), is equivalent toGibbs’ convexity criterion for the homogeneous free en-ergy, Eq. (14), which defines the classical chemical spin-odal region for mixtures without external driving. Al-ternatively, we can prove Onsager’s reciprocal relations,Lij = Lji, as a consequence of Gibbs’ maximum entropyprinciple, Eq. (11), and diffusional stability to concentra-tion fluctuations, Eq. (52). As usual in Thermodynam-ics, axioms and theorems can often be interchanged, andwhich is more fundamental is in the eye of the beholder!

With these insights, the general stability criterion (52)clearly shows that control of phase separation in a ho-mogeneous mixture results from the competition betweenauto-catalysis and chemical diffusion. Outside the spin-odal region (D > 0), a stable equilibrium system canundergo “chemical melting” (phase separation) if the re-actions are sufficiently autocatalytic (A has large enoughpositive eigenvalue). Inside the spinodal region (D < 0),a unstable mixture can undergo ”chemical freezing” (sta-bilization) if the reactions are sufficiently auto-inhibitory(A has large enough negative eigenvalue).

E. Solo-autocatalysis and Differential Resistance

The autocatalytic rate and chemical diffusion tensorscan be further decomposed to clarify the connection withequilibrium thermodynamics:

A = S −R−1G′′ (56)

D = LG′′ (57)

where we define the “solo-autocatalytic rate tensor”,

Sij =∑n

si,n∂Rn∂cj

, (58)

and the “differential reaction resistance tensor”,

R−1ij = −

∑n

si,n∂Rn∂µj

. (59)

If the reaction rates have no explicit concentration de-pendence (S = 0) and positive differential resistances(R > 0), then the Duhem-Jougeut Theorem holds: linearstability (A < 0, D > 0) requires a convex equilibriumfree energy (G′′ > 0), and the chemical spinodal range

12

remains unchanged by the driven reaction network. Inparticular, multicomponent, linear Allen-Cahn reactionkinetics (32) cannot alter the equilibrium spinodal region.Instead, the control of phase separation by reactions (inviolation of the Duhem-Jougeut Theorem) requires eithersolo-autocatalaysis (S 6= 0) or negative differential resis-tance (R < 0).

F. Nonequilibrium Gibbs Free Energy

The preceding analysis allows us to variationally con-struct the nonequilibrium free energy that determines thestability of a uniform state. In order to achieve stabil-ity in the long-wavelength limit, where reactions domi-nate diffusion (Da > 1 defined below), the constraint ofauto-inhibitory reactions, A < 0, motivates the followingdefinition, using Eq. (56):

G′′ = G′′ −RS (60)

so that stability corresponds to G′′ > 0. We see againthat unless the reaction network is solo-autocatalytic,S 6= 0, the equilibrium free energy will determine sta-bility, since G′′ = G′′, and the reactions cannot alter thespinodal region. From Eq. (60), the second variation ofG is determined by

−R−1(G′′ −G′′) = S (61)∑l,n

si,n∂Rn∂µl

δ2(G −G)

δclδcj=∑n

si,n∂Rn∂cj

(62)

which can be used to determine stability.

In some special cases, Equation (62) can be integratedto obtain the nonequilibrium free energy, or at least itsfirst variational derivative, the nonequilibrium chemicalpotential,

µnoneqi =δGδci

(63)

This is indeed possible for the simple Faradaic reactionmodel, I(c, µ) = I0(c)(µres− µ), for driven adsorption atconstant current, considered above. In that case,

∂2G∂c2

=∂2gh∂c2

+∂I∂c∂I∂µ

=∂µh∂c

+ kBT∂

∂c

I

I0

∂G∂c

= µh + (µres − µh) = µres (64)

we obtain the same nonequilibrium free energy as before,Eq. (6). The reservoir potential acts as the nonequilib-rium chemical potential of the system, µnoneq = µres,and the affinity of the reaction, A = µres − µh, is equalto the difference between the nonequilibrium and equi-librium chemical potentials.

G. Growth of Fourier Modes

The variational analysis above holds for all infinites-simal fluctuations around a time-dependent base state.Let us now consider the growth of sinusoidal perturba-

tions, i.e. Fourier modes satisfying ∇δci = ~kiδci, whichserve as a basis to represent arbitrary fluctuations. TheHessian tensor then takes the form

G′′ij = G′′ij + ~ki ·Kij~kj (65)

and the Lyapunov functional grows as,

dLcdt

=∑ij

σij

∫V

δciδcjdV (66)

where

σij = Sij −∑l

[R−1il

(G′′lj + ~kl ·Klj

~kj

)+~ki · LilG′′lj~kj + (~ki · Lil~kj)(~kl ·Klj

~kj)]

(67)

is the growth rate matrix,

∂δci∂t

=∑j

σijδcj ⇒ δ~c = eσtδ~c(t = 0) (68)

which controls the exponential growth of collective fluc-tuations.

Equation (67) expresses the general principles abovein yet another way. Since L,K > 0, regardless of equi-librium stability (signs of eigenvalues of G′′), the systemis destabilized by negative differential resistance (nega-tive eigenvalues ofR−1) or by solo-autocatalytic reactions(positive eigenvalues of S), while it is stabilized by solo-auto-inhibitory reactions (negative eigenvalues of S).

H. Negative Differential Resistance

The differential reaction resistances are usually as-sumed to be positive (like the integral resistance, R/A),but this need not be the case in electrochemistry. Themost famous example is “inverted region” of Marcus ki-netics for outer-sphere electron transfer [4, 19, 81, 82],where the differential resistance becomes negative atlarge over-potentials. The inverted region is a featureof bulk electron transfer reactions, although integrationover the Fermi distribution of electrons restores positivedifferential resistance for Faradaic reactions at metallicelectrodes [83].

To the author’s knowledge, this effect has never beenconsidered in thermodynamic stability. From Eq. (67),we see that negative differential resistance acts like back-ward diffusion with quadratic growth rate scaling as−R−1k2 as k →∞, until it is cutoff by the quartic Cahn-Hilliard gradient penalty term. At long wavelengths(k → 0), it also changes the sign of the thermodynamicterm −Rp−1G′′, which promotes stability inside and in-stability outside the equilibrium spinodal region.

13

V. APPLICATION TO DRIVEN ADSORPTION

A. Phase Field Model

Returning to the physical picture in Sec. II, let usconsider the simplest case of driven, solo-autocatalyticadsorption described by a phase-field model [4],

∂c

∂t= ∇ · L∇µ+R(c, µ, µres), µ =

δG

δc(69)

with isotropic L,K > 0. From Eq. (67), the growth rate

of the ~k Fourier mode is

σ = S −(G′′ +Kk2

) (R−1p + Lk2

)(70)

where the coefficients are all scalars:

S =∂R

∂c, R−1

p = −∂R∂µ

, G′′ =dµhdc

=d2ghdc2

. (71)

Let us analyze in detail the possibility of suppression ofphase separation (σ < 0) as the system is driven by theadsorption reaction through the spinodal region (G′′ <0), in the typical case of positive differential resistance(R−1

p > 0).The growth rate has a simple dimensionless form,

σ = S + (1− k2)(Da + k2) (72)

where the wavenumber

k2 =Kk2

|G′′|= (`k)2 (73)

is scaled to a characteristic length scale,

`2 =K

|G′′|=

κ

cskBT

∣∣∣∣dµhdc∣∣∣∣−1

(74)

which is proportional to the phase boundary thicknessand diverges at the spinodal limits (c = c/cs, µ =µ/kBT ). The growth rate and solo-autocatalytic rate

σ =Kσ

L|G′′|2= στd (75)

S =KS

L|G′′|2= Sτd (76)

are scaled to the characteristic time scale for backwarddiffusion,

τd =K

L|G′′|2=

`2

|D|(77)

where D = LG′′ < 0 is the chemical diffusivity, whichvanishes at the spinodal limits.

As usual in chemical engineering, the relative impor-tance of reactions compared to diffusion is measured bythe Damkohler number [84, 85],

Da =K

RpL|G′′|=τdτr

(78)

−S!

!σ 0

Da=0

Da=1

Da=2

k!

(a)(b)(c)

(d)

(e)

Cahn-Hilliard

Allen-Cahn

FIG. 5. Control of phase separation by driven adsorption (orelectro-autocatalysis). The band of unstable modes (orange)

can be constructed graphically from the intersection of −S =σ0 = (1 − k2)(Da + k2). For a solo-autocatalytic reaction

(a), S > 0, the instability is Allen-Cahn-like for all Da. For a

non-autocatalytic reaction (b), S = 0, the instability is Cahn-Hilliard-like for Da 1 and Allen-Cahn-like for Da 1.For a weakly auto-inhibitory reaction (c) with fast diffusion,Da 1, a narrow band of modes at finite wavelength canbe selected. For strongly auto-inhibitory reactions, (d) theinstability is Allen-Cahn-like for fast reactions, Da 1, or(e) supressed above a critical reaction rate.

which is the ratio of the diffusion and reaction time scales,only here diffusion is backward (D < 0) [18], and thecharacteristic reaction time is

τr =Rp|G′′|

=

∣∣∣∣dµhdc ∂R

∂µ

∣∣∣∣−1

(79)

which diverges at the spinodal limits (“critical slowing

down”). For a non-autocatalytic reaction, S = 0, theDamkohler number controls the shape of the growth-rate spectrum, σ0 = (1 − k2)(Da + k2), which inter-polates between the Allen-Cahn-like fast-reaction limit,σ0τr = σ0

Da ∼ 1 − k2 for Da 1, and the Cahn-

Hilliard-like fast-diffusion limit, σ0τd = σ0 ∼ (1 − k2)k2

for Da 1. Due to critical slowing down of diffusion,the Allen-Cahn-like instability dominates near the spin-odal limits (Da → 0), while the Cahn-Hilliard-like in-stability may arise only deep into the spinodal region.Such phenomena were recently studied by Lamorgese andMauri [18] for a non-autocatalytic reaction with linearAllen-Cahn kinetics [4], in which case the spinodal limitsof phase separation cannot be altered.

In contrast, control of phase separation is possible withnonlinear phase-field reaction kinetics [4]. The most un-stable wavenumber is generally given by

kmax =

√1−minDa, 1

2(80)

14

The solo-autocatalytic rate shifts the growth-rate spec-trum by a constant and selects the band of unstablemodes via σ0(k) > −S, as shown in Fig. 5.

B. Critical Rate to Suppress Phase Separation

The stability criterion, maxkσ < 0, can be expressedas a bound on the (negative) solo-autocatalytic rate,

Sτr =S

Da< − (1 + minDa, 1)2

4 minDa, 1= −F (Da) ≤ 0 (81)

or with dimensions restored,

∂R

∂c< −

∣∣∣∣dµhdc ∂R

∂µ

∣∣∣∣ F (Da) ≤ 0 (82)

where F = 1 for fast reactions (Da ≥ 1) and F ∼(4Da)−1 →∞ for slow reactions (Da 1). In the latterlimit, chemical diffusion promotes phase separation, so anincreasingly negative (auto-inhibitory) solo-autocatalyticrate is required to maintain stability.

At constant reservoir potential, Equation (82) impliesan upper bound on the total autocatalytic rate,

A =

(dR

dc

)µres

=∂R

∂c+∂R

∂µ

dµhdc

< −∣∣∣∣dµhdc ∂R

∂µ

∣∣∣∣ (1−minDa, 1)2

4 minDa, 1(83)

For fast reactions, Da > 1, we recover the constant-potential stability criterion, A < 0, discussed in SectionII, Eq. (3).

At constant current, the stability criterion can be ex-pressed as a lower bound on the derivative of reservoirpotential with respect to reaction extent (or time),(

∂µres∂c

)R,c

>RrRp

∣∣∣∣dµhdc∣∣∣∣ (1−minDa, 1)2

4 minDa, 1(84)

where we define the reactant differential reaction resis-tance,

R−1r =

∂R

∂µres(85)

which is positive for driven adsorption on interfaces orelectrodes. The reactant and product differential resis-tances are equal for Carati-Lefever kinetics (35), whichincludes the limit of linear kinetics (31), but for our moregeneral model (37), which includes Butler-Volmer kinet-ics [4], they are typically different, Rp 6= Rr. For fastreactions, Da > 1, Equation (84) reduces to the constant-current stability criterion, ∂R

∂µres> 0, discussed in Section

II, Eq. (4).

C. Control of Phase Separation byElectro-autocatalysis

Finally, we are ready to apply our general stability the-ory to electro-autocatalysis. Consider generalized Butler-Volmer kinetics, Eq. (39), for symmetric charge transfer

(α = 12 ) as a model for cation reduction and adsorption

or intercalation from an electrolyte reservoir to a cathodesurface [4]:

I = neR = 2I0 sinh

(µres − µ

2

)(86)

where the exchange current density

I0 = Ir(1− c)eµ/2 (87)

makes the reaction solo-auto-inhibitory (S < 0) andthus capable of suppressing phase separation, as a re-sult of lattice crowding, γ‡ = (1 − c)−1. The prefactor,Ir = nek0

√aresae, is constant, if we assume constant

chemical activities of the electrolyte and electrons (fixedband structure). The effective reservoir chemical poten-tial, µres, is then controlled by the cathode potential,since the activation over-potential is η = (µ − µres)/ne.Assuming that the anode is held at constant potential,the cell voltage is V = V0−neµres, where V0 is the opencircuit voltage when µres = µ = 0.

Using

∂I

∂c= − I

1− c,

∂I

∂µ=I

2−

√I20 +

(I

2

)2

(88)

the stability criterion, Eq. (82), implies

I>2Ir(1− c)eµh/2√[

1 + 2(

(1− c)∣∣∣dµh

dc

∣∣∣F (Da))−1

]2

− 1

(89)

This is an implicit equation for the (positive) criticalcurrent Ic(c) that suppresses phase separation, since theDamkohler number is current-dependent:

Da =K(√

(2I0)2 + I2 − I)

2nekBT |D|(90)

according to Eqs. (78) and (88).

D. Role of Diffusion

There are two different regimes of stability, dependingon the importance of diffusion compared to reactions, asdefined by the Damkohler number:

• Slow diffusion. For relatively fast reactions (Da ≥1), the critical current is given by the bound in Eq.(89) with F = 1. Interestingly, the stability crite-rion is independent of the diffusivity for all Da ≥ 1,not only in the asymptotic limit of slow diffusion(Da 1).

15

A

B

C critical current

(a) (b)

(d)

(c)X=20% X=50% X=90%

!I

FIG. 6. Electro-autocatalytic control of phase separation in an Allen-Cahn Reaction model for lithium insertion in LFP [14],based on phase-field Butler-Volmer kinetics with regular solution thermodynamics [4] (neglecting coherency strain [7], B =0). (a) Sketch of the insertion reaction and depth-average concentration. (b) Simulations of the concentration profile and(c) cell voltage (versus a constant lithium reference) for three different currents, which are indicated as linear paths in the

(d) “nonequilibrium phase diagram” of thermodynamic stability, I > Ic(c), in the plane of applied current I = I/Ir andhomogeneous concentration X = c. [Adapted from Bai et al. [14]]

• Fast diffusion. Once the diffusivity surpasses acritical value defined by Da > 1, the critical cur-rent increases. Destabilizing chemical diffusionthen begins to dominate over stabilizing electro-catalysis. In the limit of fast diffusion (Da 1,F ∼ (4Da)−1), the critical current has the asymp-totic form

Ic(c) ∼(nekBTD

4K

)c(1− c)

(dµhdc

)2

=necs(1− c)

4τd(91)

which scales with the diffusion current, necs/τd(full capacity per diffusion time). Although thecritical current does not depend on rate-constantprefactor, Ir, in this limit, it still depends onthe concentration-dependence of the reaction rate(electro-autocatalysis). Indeed, the general stabil-ity criterion (82) for Da 1 can still be expressedas a bound on solo-auto-inhibition

∂R

∂c< −|D|

4K

∣∣∣∣dµhdc∣∣∣∣ (92)

which takes the form of Eq. (91) using Eq. (88).

E. Regular Solution Thermodynamics

The critical current, Ic(c,Da), separates the stable andunstable regions of the “non-equilibrium phase diagram”of current versus concentration (and temperature), whichare traversed during the dynamics. An example is shownin Fig. 6(d) for Butler-Volmer kinetics with regular so-lution thermodynamics [4, 14]:

µh = lnc

1− c+ Ω(1− 2c) + B(c−X) (93)

dµhdc

=1

c(1− c)− 2Ω + B (94)

where Ω is the enthalpy of mixing particles and vacan-cies. For solid-state intercalation, the last term derivesfrom the elastic coherency strain energy for small fluctu-ations [7, 26], where X is the average concentration, andc = X for a homogeneous base state. Without strain(B = 0), equilibrium in this model (µ =constant) cor-responds to the Frumkin isotherm for adsorption withlateral forces [86].

For this reaction model, Figure 6 shows simulations ofthe Allen-Cahn Reaction equation, Eq. (69) with L = 0,which confirm the predictions of the stability theory [14].The concentration profiles in (b) develop long-wavelengthfluctuations (set by the geometry) which grow as the sys-

16

Quantitative Evidence (2015) 1. Low-rate Li Insertion (0.6C) 2. Low-rate Li extraction (-0.6C) 3. High-rate Li insertion (2C)

61 71 54 12 30 61 78 86 2 14 17 22 27 31

!I0 !I

unstable

stable

!c !c c

Exchange current: Experiment Exchange current: Theory

Predicted Stability Diagram (a) (b) (c)

(d)

Experiment (2016)

Phase-field BV (2011)

Classical BV (1993)

stable

1

2

3

fast reactions slow reactions slow reactions fast reactions fast reactions

fast slow

FIG. 7. Direct experimental evidence for the control of phase separation by electro-autocatalysis in single nanoparticles oflithium iron phosphate, obtained by in operando scanning transmission x-ray microscopy (STXM) [15]. (a) Exchange currentversus local surface concentration, obtained by pixel-level analysis of STXM movies of lithium evolution, (b) compared with theexchange current for Butler-Volmer kinetics from our original phase-field model [14] and traditional porous electrode theory [21].(c) Predicted linear stability diagram versus current and composition for the exchange current curves in (b) from models andexperiment. (d) Typical sequence of STXM images showing the lithium concentration profile (X = 0, 0.5, 1.0 in LiXFePO4 forgreen, yellow, red) in a ∼ 1µm sized platelet particle (150nm thick in the depth direction) during cycling at different rates. Ata moderate insertion rate (0.6C= 100min to full capacity) some some phase separation occurs, which is enhanced significantlyduring extraction at the same rate (-0.6C). Next, at a high current (2C= 30min discharge) above the critical insertion rate,phase separation is suppressed (“electrochemical freezing”), and uniform, stable insertion is observed. [Adapted from Limet al. [15]]

tem passes through the unstable region of nonequilibriumphase diagram (d), as signified by of increasing battery

voltage in (c) (dVdc ∼ −dµres

dc > 0). The fluctuations de-cay as soon as the system re-enters a stable region of (d),and the voltage begins to decrease again in (c).

The control of phase separation is demonstrated bythree currents in Fig. 6: (A) For small currents, I Ic,the system overshoots the phase-separated equilibriumvoltage plateau, undergoes spinodal decomposition, andthen closely follows the voltage plateau, offset only by asmall activation overpotential, associated with the mov-ing phase boundary, or “intercalation wave” [14, 85]. (B)At larger currents, I < Ic, the instability is hindered,and the system behaves as a “quasi-solid solution” in theunstable regions of increasing voltage. (C) Above thecritical current, I > Ic, the homogeneous solid solutionis stable, and the voltage decreases monotonically as aresult of the concentration-dependent activation overpo-tential (yellow arrows in (c)).

VI. EXPERIMENTAL EVIDENCE

A. Lithium Iron Phosphate Intercalation Kinetics

Strong experimental support for the present theory hasrecently been achieved, after a decade of research in thefield of Li-ion batteries. Many battery materials exhibitmultiple phases with varying composition, voltage, andtemperature [87–89], and our nonlinear phase-field reac-tion model, Eqs. (37) and (40), was first developed forthis application, starting in 2007 [4, 85]. In the prototypi-cal case of lithium iron phosphate (LFP), the model led tothe surprising prediction that insertion reactions can sup-press phase separation in nanoparticles above a criticalcurrent [14], even in the presence of heterogeneous nucle-ation [14, 90] and elastic coherency strain [7], althoughthe underlying mechanism – electro-autoinhibition – wasnot explained until now.

This theoretical prediction helped to explain the dra-matic reversal of fortune of LFP as a battery material. Inthe original paper on LFP, Goodenough and co-workers

17

concluded that “this material is very good for low-powerapplications; at higher current densities there is a re-versible decrease in capacity that, we suggest, is associ-ated with the movement of a two-phase interface”[91]. In-deed, phase separation is undesirable since it damages thecrystal with coherency strain and lowers the rate capabil-ity by storing lithium in non-reactive stable phases [4, 14].Within a few years, however, LFP was reformulated asnanoparticles [88] with conductive coatings and demon-strated ultrafast (< 10 sec) discharge without clear signsof phase separation in the voltage profile [92], despitethe assumption of two-phase “shrinking core” particlesin prevailing mathematical models [93, 94].

The new theory was controversial, however, and com-peting hypotheses were made. The existence of a “solidsolution pathway” of uniform insertion (and extraction)in LFP was suggested, on the basis that classical nucle-ation theory would prohibit nucleation and growth [95].On the other hand, our phase-field model predicted thatphase separation can nucleate at surfaces and collapsedexperimental data for the size-dependent nucleation bar-rier [90]. Phase separation was later observed in situ inLFP porous electrodes [96], and compared with phase-field porous electrode simulations [97, 98].

In 2014, three groups reported the first experimentalevidence for the suppression of phase separation in LFPat high insertion rates [99–101], although none could set-tle the question of the mechanism. Zhang et al. [99] andLiu et al. [101] used in situ synchrotron diffraction tomeasure the volume averaged Li+ site occupation dis-tribution (Fe+3/Fe+2 redox state). Each study founda transition from two-phase to solid-solution transfor-mation above a critical current [99, 101] but could notobserve the concentration profiles or reaction kinetics.Meanwhile, Niu et al. [100] were the first to directlyobserve nonequilibrium solutions in LFP nanowires, al-though the situation was artificial and could not shedlight on the reason for their stabilization.

In 2016, Lim et al. [15] achieved a remarkable first testof the theory by in operando scanning transmission x-raymicroscopy of single LFP nanoparticles in a microfluidicelectrochemical cell. The two-dimensional lithium con-centration evolution was directly observed with nanoscaleresolution over the active facet of platelet-like nanopar-ticles, during realistic conditions of battery cycling. Themassive dataset of pixels from many movies of concen-tration evolution allowed the team to extract the lo-cal current density, and hence the exchange rate, ver-sus local concentration, and the experimental curve (Fig.7(a)) is asymmetric and similar to the original phase-field model of Butler-Volmer kinetics [4, 7, 14], and differ-

ent from the symmetric form, I0 ∼√c(1− c), assumed

in traditional diffusion models (Fig. 7(b)). The ex-perimental and phase-field insertion reactions are solo-autoinhibitory across the spindoal region, which leads tosuppression of phase separation above a critical insertioncurrent and enhanced instability during extraction (Fig.7(c)), as explained in Fig. 1. In contrast, the symmetric

reaction model predicts phase separation at all currents.As shown in Fig. 7(d), the data for repeated cycling

of single nanoparticles confirm the theoretical predictionfor the asymmetric exchange current. Lithium insertionat a moderate rate (0.6C=100 min. discharge) exhibitsquasi-solid solution behavior with non-uniform concen-tration, while extraction at the same rate produces clearphase separation. Re-insertion in the same nanoparti-cle at a higher rate (2C=capacity in 30 min. discharge)leads to stable, uniform filling, but high-rate extraction(not shown) still leads to phase separation. When thecurrent is turned off at intermediate concentrations (notshown), spinodal decomposition leads to striped equilib-rium phase patterns, also predicted by the model withcoherency strain [7]. Previous models [20, 21] could notpredict any of these observations.

B. Lithium Peroxide Electrodeposition Kinetics

Our general theory can also be applied to epitaxial sur-face growth or electrodeposition, where the surface heighth(x, t) acts as a surface concentration c(x, t) integratedover the depth of the deposit [102]. In that case, the freeenergy functional G[h] contains different physical effects,such as orientation-dependent surface energy and discretestable monolayers, but the reaction kinetics can still bedescribed by phase-field Butler-Volmer kinetics. In thiscontext, the instability of a uniformly growing film to“phase separation” corresponds to the homogeneous nu-cleation and growth of islands, which can be controlledby electro-autocatalysis, according to the same principlesrevealed by studies of ion intercalation.

In the case of lithium peroxide deposition in Li-air bat-tery cathodes, the model successfully predicted a transi-tion from island growth at low rates to homogeneous,random deposition at moderate rates, in good agreementwith experimental voltage profiles and ex situ observedgrowth morphologies [102]. This is the surface-growthanalog of suppression of phase separation in driven ad-sorption. Although the local current density and ex-change current could not be measured, this experimentshows the generality of the present theory, which is byno means limited to adsorption phenomena. Despite thescientific interest of this result, however, it also revealsa fatal flaw for Li-air batteries, since thick uniform filmsof lithium peroxide block electron transfer and lead toinefficient battery cycling [103].

VII. OUTLOOK

A. Solid State Ionics

Li-ion Batteries. This work opens the possibility ofdesigning interfaces of intercalation materials to controlphase separation [4, 15], as well as structural phase tran-sitions at electrochemical interfaces [104]. A key goal to

18

400 nm

400 nm

(b)

(c)

(a)

(a)

FIG. 8. Control of pattern formation by electro-autocatalysisfor lithium peroxide electrodeposition in Li-air battery cath-odes [102], predicted by the same general theory. (a) Phase-field simulations of surface height evolution driven by gen-eralized Butler-Volmer kinetics, which capture the observedvoltage behavior (not shown) and morphological transitionswith increasing current. Homogeneous nucleation and growthof islands at very low currents leads to the experimentallyobserved disk-like particles of Li2O2 shown in (b) on car-bon nanotube current collectors. At larger currents, thegrowth becomes more random and ultimately uniform layer-by-layer above the predicted critical current, as observed in(c). [Adapted from Horstmann et al. [102].]

improve the rate capability and lifetime of Li-ion batter-ies is to suppress phase separation during ion insertion,which can be the rate limiting step for both dischargingand re-charging of the battery, corresponding to ion in-sertion at the cathode or anode, respectively. In contrast,ion extraction at the opposite electrode tends to requireless overpotential and might not be affected as much byphase separation. At either electrode, surface phase sep-aration reduces the active area (to that of the exposedphase boundary) and causes degradation via mechanicaldeformation and side reactions, such as Li metal platingat the anode during recharging (leading to capacity fadeand safety hazards), which become favored once surfaceion concentrations reach stable phases.

The theory provides some guidance for surface mod-ification of the active materials to achieve these goals.For generalized Butler-Volmer kinetics (39), the solo-autocatalytic rate is related to the transition state ac-

tivity coefficient, S ∝ ∂γ−1‡∂c , which can be altered by

blocking sites to reduce the configurational entropy or de-positing coatings that cause attractive or repulsive forceswith intercalated ions to adjust the enthalpy. These ef-fects may play a role in the observed (but poorly un-derstood) rate-enhancing effects of phosphate or othersurficial glass films on LFP and other cathode materialsfor Li-ion batteries [92, 105].

These ideas can be coupled with existing strategies toalter surface chemistry. For example, Park et al. [106]showed that anion surface modification of LFP by nitro-gen or sulfur adsorption improves the insertion rate ca-

pability, which they attributed lowering of the barrier forlithium ion insertion (µ‡ = kBT lnγ‡) by stabilizing theunder-coordinated Fe2+/Fe3+ redox couple at the sur-face. This chemical bonding effect should be stronger atlow lithium concentrations (as in their ab initio calcula-tions [106]) due to the lower conductivity of FePO4 limit-ing access of electrons to the redox site. In that case, ourtheory predicts that if the exchange current were prefer-entially enhanced at low concentrations, then the reac-tion would become more solo-autoinhibitory across thespinodal region, thus further suppressing phase separa-tion and contributing to the observed rate enhancement.

Resistive Switching Memory. Electro-autocatalysismay also find applications in the forming of redox-basedresistive random access memories (ReRAM) [107], whichare non-volatile, low-power alternatives to today’s flashmemory. Promising examples include Valence ChangeMemories (VCM), based on the controlled dielectricbreakdown of transition metal oxide thin films [108].In the forming cycle at high voltage, metal interstitialsor oxygen vacancies undergo compositional instabilitiesto form conducting filaments of valence-changed metalcations, which are then used to reversibly short circuitthe film as a means of information storage. In thickfilms of perovskite titanates, the forming step has beenobserved as a bulk fingering instability of the “virtualcathode” of condensed oxygen vacancies [109, 110], butin ultra-thin films, Faradaic surface reactions may playa more dominant role. By tuning the solo-autocatalyticelectron transfer rate at the cathode, e.g. by modifyingthe surface charge and local cation valence as above, itmay be possible to control the most unstable wavelengthof the instability during forming, and thus the size andspacing of the conducting filaments.

Hydrogen storage. These effects are not limited to elec-trodes but also apply to the intercalation materials forneutral species. Hydrogen insertion in silicon or palla-dium nanoparticles has been explored for hydrogen stor-age and also arises in catalytic materials. Binary phaseseparation in PdH nanoparticles has been observed insitu and manipulated via the hydrogen gas pressure [11–13]. It would be interesting to study the response to sud-den, large gas pressure steps to see if phase separationcan be suppressed, leading to faster, more uniform in-tercalation. Similar surface modification strategies couldalso be used to manipulate driven autocatalysis.

B. Electrokinetics at Liquid Interfaces

Electrovariable Nanoplasmonics. A major theme ofthis Discussion is electrovariable optics, based on the re-versible deposition of plasmonic nanoparticles at immis-cible liquid interfaces driven by electric fields [111, 112].The theory of electrovariable nanoplasmonics focuses onthe effective trapping potential in the normal direction tothe interface [113] and includes a thermodynamic modelfor nanoparticle adsorption and deposition kinetics[114].

19

The model focuses on the repulsive electrostatic forcesbetween adsorbed nanoparticles, which have the samecharge and polarization in the applied field, and thuspredicts a stable uniform monolayer, amenable to fastswitching.

Although electrostatic repulsion may dominate, thereare other strong forces at the nanoscale that could leadto lateral attraction and thermodynamic instability of ananoparticle monolayer, as shown in Fig. 9(a). De-pending on contract angles, electrostatic forces and tran-sient geometrical constraints, attractive capillary “dim-ple” forces can be very strong for nano-menisci and couldlead to clustering (the “cheerios effect”[115]). More-over, attractive entropic depletion forces can be tunedby adding surfactants to the system, which would causeinter-particle attraction to reduce the excluded volumefor surfactants.

In the presence of attractive lateral forces, the responseto an applied electric field becomes more interesting. Ifthe system tends to phase separate into clusters at theinterface, then the present theory predicts that electro-catalytic adsorption reactions, e.g. obeying generalizedButler-Volmer kinetics (39) with regular solution thermo-dynamics (93), would stabilize the interfacial monolayerduring deposition (Fig. 9(b)) and destabilize it duringdesorption (Fig. 9(c)), if the adsorption reaction is auto-inhibitory, and vice versa, if it is autocatalytic. Stableuniform deposition should proceed more quickly than un-stable cluster dissolution, due to the larger active area [4].This prediction should be tested experimentally and anypatterns characterized, especially with enhanced attrac-tive lateral forces. Since clustering transitions on theinterface affect optical properties, it may be possible toexploit these phenomena in new device designs. For ex-ample, switching between a clustered state and uniformcoverage without significant mass transfer from the bulksolution could enable faster switching with tolerable res-olution.

Electrophoretic Displays. Similar issues arises inelectrophoretic displays or “electronic paper”, where col-loidal pigments or particles are shuttled between trans-parent electrodes in liquid-droplet pixels [116]. It is wellknown that colloidal dispersion stability is important inthe bulk liquid, but there is also ordering on the surfacethat can interfere with device operation [117]. This clus-tering contributes to the inability of electronic paper toswitch fast enough to enable the playing of movies, andperhaps it could be better understood or even controlledusing the ideas in this paper.

Ionic Liquids and Solids. Room temperature ionicliquids exhibit complex charge oscillations at electrifiedinterfaces [118]. To some extent, these phenomena canbe understood in terms of ion crowding [119] and over-screening due to strong Coulomb correlations [120, 121],but recent models have also included additional short-range forces that improve predictions of double-layercapacitance [122] and promote the “phase separation”of like-charged domains [40], in the hope of explaining

+ + + + + + + +

+ +

+ +

qE

Fatt

Fast uniform adsorption Slow cluster desorption

(a)

(b) (c)

FIG. 9. Application of the theory to electrovariable opticswith plasmonic nanoparticles adsorbing on an immiscible liq-uid interface. (a) Attractive lateral forces Fatt (pink) can re-sult from depletion forces of surfactants or capillary forces me-diated by nano-menisci and compete with adsorption drivenby the normal electric field. (b) For auto-inhibitory insertion,phase separation can be suppressed (b) above a critical rate,leading to fast, uniform insertion, but in that case, (c) thereverse autocatalytic extraction would destabilize the mono-layer and promote phase separation, leading to slow interfacialextraction of clusters.

long-range charge oscillations [123, 124] and other pat-terns [118, 125].

The treatment of attractive short-range forces and lat-tice repulsion in these models [40, 122] is similar toCahn’s regular solution model for binary solid alloys [2],also considered here for solid-state ion intercalation [4],Eq. (93), although electrostatic interactions are alsoconsidered. As such, the principles of electroautocatal-ysis and clustering described here might be useful inunderstanding the switching dynamics and ordering ofions in applications to electric double layer capacitors.Moreover, the coupling between double layer structureand Faradaic reactions may be important in understand-ing the large electrochemical window of “solvent-in-salt”ionic-liquid-based electrolytes for rechargeable batter-ies [126, 127].

It should be emphasized that our analysis here doesnot explicitly consider electric fields from diffuse chargeor other long-range forces. The stability analysis is basedon a homogeneous free energy for short range forces plusa gradient correction, and the focus is on electrochemi-cal reactions in neutral electrolytes with negligible dou-ble layer effects. Past mean-field theories of solid elec-trolytes [128] and defect dynamics [129] accounting forinteractions between space charge and Butler-Volmer ki-netics have not found any unusual effects on phase sep-aration, so care must be take in applying our results tocharged systems.

20

C. Patterns Driven by Electron TransferReactions

Electron Transfer in Solution. Electron transfer re-actions between donor and acceptor atoms have mostlybeen studied by chemists at the molecular level withoutconsidering how quantum mechanical effects might in-fluence macroscopic reaction-diffusion phenomena. Anintriguing new prediction of our theory is the destabi-lizing effect of negative differential reaction resistance,which is the defining characteristic of the “inverted re-gion” of Marcus kinetics for outer-sphere electron trans-fer [4, 19, 130]. It is interesting to note that it took severaldecades after the pioneering work of Marcus [81, 82], untilMiller, Calcaterra and Closs [131, 132] managed to ob-serve the predicted effect of exothermocity (∆rG) on thekinetics of intramolecular electron transfer, after manyinconclusive studies on inter-molecular electron transfer.

Our results suggest that thermodynamic instabilityof reactive electrolytes in the inverted region of inter-molecular electron transfer could have played a role inthese experimental challenges, due to the coupling of re-actions with rapid density fluctuations. In order to testthis prediction, it would be interesting to revisit the orig-inal experiments [131, 132], by measuring density fluctu-ations of the redox species (e.g. by x-ray or neutron ad-sorption spectroscopy) following a pulse of solvated elec-trons in a reactive liquid electrolyte (e.g. biphenyl an-ions in acceptor organic solvents). Using combinationsof intra- and inter-molecular electron transfer, it maybe possible to use our theory to control the instabilityto achieve new types of nanoscale patterns for materialssynthesis or actuation.

Electron Transfer at Electrodes. Perhaps for sim-ilar reasons, it took just as long after Marcus’ theoryof electron transfer on electrodes [133] before it was firstverified experimentally by Chidsey [134], again for intra-molecular electron transfer, across self-assembled mono-layers. Recently, the first evidence of Marcus-Hush-Chidsey kinetics was reported for solid-solid electrontransfer in porous electrodes of Li-ion batteries, wherethe rate-limiting step was attributed to electron transferbetween the iron redox site in LFP and the carbon coat-ing of the active particles [135]. While these experimentsshowed the importance of coupling electron transfer reac-tions with compositional dynamics, however, they did notprobe the effects of negative differential resistance, sinceintegration over the Fermi distribution of electrons elimi-nates the inverted region for a metallic electrode [83, 136].

Last year, the Marcus inverted region was observed forthe first time on a semiconductor photo-electrode[137],which, according to our theory, could lead to novel insta-bilities and pattern formation in photo-electrochemicaldevices. Yet again, the experiments involved intra-molecular electron transfer, from single-walled carbonnanotubes to acceptor molecules in fullerene derivatives.Besides photo-electrochemistry and photovoltaics, func-

tionalized carbon nanotubes and graphene sheets are alsoincreasingly used for dynamical processes, such as ther-mopower waves for electrical energy generation [138],where our theory could shed light on the stability of heatand mass transfer coupled with electron transfer reac-tions.

D. Electro-autocatalytic Control ofMicrostructure

Hydration and Precipitation of Cement Paste. One ofthe most important examples of electrochemical patternformation is the hydration of tricalcium silicate (the mainmineral component of portland cement) and precipitationof calcium-silica-hydrate (C-S-H) gels to form cementpaste [139, 140]. This multi-step reaction is known tobe autocatalytic [140]. Microstructural evolution, lead-ing to the unique strength of cement paste, proceeds byspinodal decomposition of precipitating C-S-H particles,as shown in recent molecular simulations [141, 142].

The hardening of cement paste is a natural candi-date for continuum modeling with our reactive phase-field model, Eqs. (37) and (40). Our general stabilityanalysis may help explain how electroautocatalytic pre-cipitation reactions drive spinodal decomposition to de-termine the final microstructure. The theory may alsoprovide insights into how the paste microstructure couldbe controlled by varying the initial mixture compositionor by applying electrical current during curing, since elec-tric fields are already known to induce microstructuralchanges in hardened cement [143]. More generally, mod-els of driven precipitation may find applications in otherelectrochemical systems, such as aqueous Li-air batter-ies [144].

Electrodeposition. We have already discussedhow electro-autocatalysis enables morphological con-trol of lithium peroxide electrodeposits [102], andthere are many other possible applications in electro-deposition/dissolution. For example, nanostructuredredox-polymer electrodes for super capacitors have beenmade by simultaneous electropolymerization of pyrroleand electro-precipitation of polyvinylferrocene [145], andthe microstructure depends on the applied current andelectrolyte composition.

Corrosion. A more direct application of our theoryarises in the corrosion of binary metallic solids [146, 147].The de-alloying of a Ag/Au solid solution by selective dis-solution of the more electrochemically active metal (Ag)can leave behind three-dimensional nano-porous struc-tures of the more noble metal (Au), which result frommodulation of the corrosion rate by surface spinodal de-composition. It was found that simulations based on theregular-solution Cahn-Hilliard equation could reproducethe experimental microstructure only for a certain choiceof the concentration-dependent exchange current [146],I0 ∝ e−c/c

∗, rather than the usual assumption of mass

action kinetics, proportional to the silver adatom con-centration, I0 ∝ (1 − c). In hindsight, this is yet an-

21

other phenomenon of phase-separation control by electro-autocatalysis, which could be tailored to achieve a desiredpore size guided by our theory.

E. Control of Phase Separation in Biology

Bacteria and Active Matter. Here we have focused onopen chemical systems driven by externally controlled re-actions, but there are other types of driving work thatcould fit into our general theoretical framework. For ex-ample, active Brownian suspensions of swimming parti-cles, such as E Coli bacteria, exert “swimming stress”on their surroundings [76], which leads to phase sepa-ration that can be described by a nonequilibrium freeenergy [76–80]. Equation (49) implies that, at least neara nonequilbrium steady state, active diffusion is gener-ally destabilizing, while passive diffusion is stabilizing.Introducing reactions among active swimmers with envi-ronmental chemicals could provide an interesting meansof tuning their pattern formation.

Protein Phase Separation in Cells. Over the pastdecade, there has been growing appreciation of phaseseparation in biology [148, 149], stimulated by the dis-covery of liquid-liquid protein phase separation insidethe cytoplasm of embryonic cells, leading to cell divi-sion [150]. Although it is known that reactions such asRNA/protein binding play a role in controlling phase sep-aration [151], most experiments and models have focused

either on applied protein concentration gradients with-out reactions [152] or on the evolution of already formeddroplets [153], regulated by autocatalytic reactions [154],including suppression of Ostwald ripening [155].

The present theory could be useful in understandinghow driven autocatalytic reactions can stabilize the ho-mogeneous mixture or trigger the onset phase separa-tion and control the nascent patterns the lead to liquidorganelles. The general notion of pattern formation bychemically driven phase separation has a long history inbiology relating to the origins of life [153]. This possibil-ity also fascinated Prigogine [156] and motivated muchof his own work in nonequilibrium thermodynamics [5].

ACKNOWLEDGMENTS

This work began during a sabbatical leave at StanfordUniversity supported by the Global Climate and EnergyProject and by the US Department of Energy, Basic En-ergy Sciences through the SUNCAT Center for InterfaceScience and Catalysis. The author is grateful to Peng Baiand Yiyang Li for help with the figures and insights frombattery simulations, Dimitrios Fraggedakis for checkingthe calculations and noting the second term in Eq. (52),and David Zwicker, Sho Takatori, Thomas Petersen, Al-bert Tianxiang Liu and Dimitrios for valuable references.

[1] This invited paper will be published in a special issue ofFaraday Discussions for Chemical Physics of Electroac-tive Materials, April 10-12, 2017, Cambridge, UK.

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